1. Field of the Invention
Examples of the subject matter disclosed herein generally relate to methods and systems for seismic exploration and, in particular methods and systems for processing and analysis of seismic data using constrained least-squares spectral analysis.
2. Description of the Prior Art
History
Spectral decomposition of reflection seismograms was introduced as a seismic interpretation technique by Partyka (1999). He recognized that seismic frequency spectra using short windows were greatly affected by local reflectivity spectra, and thus carried information about layer characteristics. He showed that simple layers of certain thicknesses exhibit notched spectra, and that the pattern of frequencies at which these notches occur can sometimes be used to infer layer thickness. He also showed that, for this reason, seismic images at different frequencies preferentially illuminate, or respond to, geological variations differently. Spectral time-frequency analysis has since become an important practical seismic interpretation tool that has achieved widespread use.
Early spectral decomposition work primarily used (1) the Short Time Fourier Transform (STFT), which is equivalent to the cross-correlation of the seismic trace with a sinusoidal basis over a moving time window, (2) the Continuous Wavelet Transform (CWT), which is the cross-correlation of the seismic trace against a wavelet dictionary, and (3) Matching Pursuit Decomposition (MPD), which is the decomposition of the seismic trace into basis atoms. The use of these methods for seismic time-frequency analysis is discussed by Chakraborty and Okaya (1995).
The literature is rich in papers discussing geological applications of seismic spectral decomposition, a few of which are mentioned here. The STFT has been successfully applied for stratigraphic and structural visualization (e.g., Partyka, et al., 1999; Marfurt and Kirlin, 2001). Marfurt and Kirlin (2001) derive a suite of attributes, including peak frequency, from spectral decomposition volumes in order to efficiently map stratigraphic features, particularly fluvial channels. These frequency attributes are further described and applied by Liu and Marfurt (2007). Sinha et al. (2005) apply the CWT for stratigraphic visualization and direct hydrocarbon indication. Matos et al. (2010) compute CWT spectral decomposition phase residues as an attribute for stratigraphic interpretation. Castagna et al. (2003) and Fahmy (2008) use MPD for direct hydrocarbon detection. Partyka (2005), Puryear (2006) and Puryear and Castagna (2008) describe the use of spectral decomposition as a driver for thin-layer reflectivity inversion.
Higher resolution seismic spectral decomposition methods would assist in the interpretation of geological features masked by spectral smearing (when the STFT is used) or poor temporal resolution at low frequencies (when the CWT is used). Toward this end, we revisit Fourier theory and then formulate an alternative approach to seismic spectral decomposition using Constrained Least Squares Spectral Analysis, which potentially has advantages over conventional methods such as the STFT and CWT in terms of improved temporal and/or frequency resolution of seismic reflection data.
Introduction
Seismic spectral decomposition (e.g., Partyka. et al., 1999) transforms each reflection seismogram into a time-frequency space that represents localized frequency content as a function of seismic record time. Thus, individual seismic volumes are transformed into multiple frequency volumes that preferentially highlight geophysical responses that appear within particular frequency bands. Commonly used spectral decomposition methods, such as the Fourier Transform and the Continuous Wavelet Transform generally require a tradeoff between time and frequency resolution that may render them ineffective in particular cases for certain interpretation applications, such as layer thickness determination and direct hydrocarbon detection. The objective of this paper is to introduce and evaluate the effectiveness of Constrained Least Squares Spectral Analysis (CLSSA) as a seismic spectral decomposition method and show that it has resolution advantages over the conventional approaches.
Fourier-based spectral decomposition uses a sliding temporal window, which limits both temporal and frequency resolutions. In spectral analysis of seismic events that are near in time to other arrivals, it is often necessary to sacrifice frequency resolution by using a short time window to isolate the event of interest. FIG. 1 illustrates this fundamental problem in spectral decomposition: A pair of reflection coefficients from the top and base of a thin layer is bracketed by nearby strong reflection coefficients. When convolved with a wavelet, the reflection event from the thin layer has interference at its fringes with side-lobes from the bracketing reflections. The correct spectral response for the thin layer (a cosine function times the Ricker spectrum) should have a notch at 50 Hz corresponding to the first spectral notch of the even impulse pair. As emphasized by Partyka (1999), the frequency at which this notch occurs could be used to determine the layer time thickness, which is of great potential utility for pre-drill estimates of reservoir volumetrics. In order to make use of the notch occurrence, the window chosen for spectral analysis must be short enough to avoid interference with nearby reflectors, but long enough so that window smearing effect on the spectrum does not change the notch location. Unfortunately, as illustrated in FIG. 1, this may not be achievable in practice. A Hann window short enough to avoid interference (40 ms in this case) results in a Fourier spectrum that is dominated by the window spectrum, and the notch and peak frequencies do not directly reflect the layer characteristics—which would thereby yield an incorrect reservoir thickness estimate. Longer windows yield spectral estimates that are corrupted by the interfering energy and again yield misleading spectral notch frequencies that would result in incorrect thickness estimates. In such a situation, it would be advantageous to be able to use a short-window without the corresponding loss of frequency resolution inherent in the use of the Fourier Transform.
Short windows are desirable for the temporal isolation of particular portions of seismic traces in order to obtain spectra and spectral attributes (such as peak frequency and amplitude at peak frequency) that are relevant to the characteristics of a given layer. However, the Fourier Similarity Theorem (e.g., Bracewell, 1986) requires that shorter windows of a given shape have poorer frequency resolution which can mask and modify spectral characteristics. Using a given window shape, better frequency resolution can only be achieved with the Fourier Transform at the expense of poorer time resolution by increasing the window length. Reducing the window effect in seismic time-frequency analysis is, thus, of great potential practical significance.
One approach towards reducing the window effect is to circumvent the Fourier Transform, and solve directly for the Fourier Series coefficients using least-squares analysis within a window (Vaniek, 1969). The Fourier Transform is indeed the least-squares solution for the Fourier Series coefficients, only when the sinusoidal basis functions are orthogonal. When seismic data are windowed, this definition is violated for those frequencies for which the window length is not an integer number of periods. The well-known consequence is smearing of the data spectrum computed with the Fourier Transform by the window transfer function. However, this effect is a result of the definition of the Fourier Transform requiring that the sinusoidal bases are uncorrelated, not a necessary consequence of Fourier Analysis (which is the determination of the Fourier Series coefficients). The Fourier Transform is only one of many possible means of solving for those coefficients. From the point of view of determining the Fourier Series coefficients of a time series within a window, the window smearing effect arises from what can be considered the incorrect implicit requirement of the windowed Fourier Transform that the sinusoidal bases are orthogonal over the window length. This results in the Fourier Transform yielding the spectrum of the windowed data rather than the spectrum of the data within the window. Seismic time-frequency analysis by direct solution of the normal equations for the Fourier Series coefficients when the sinusoidal bases are not orthogonal has, perhaps surprisingly, not been reported upon in the seismic spectral decomposition literature. Such an approach is complicated by the fact that the inversion for these coefficients is non-unique, and constraints are thus required.